Rarely-interacting atoms are straightforward to model thermodynamically. That these atoms rarely interact is the
basic premise of the ideal gas law and the ideal solution approximation. For example, theBoltzmann approximation
for ions in an electrolyte consists of treating the ions as if they do not interact with each other, but rather a
continuum electrical field. The solvent molecules (typically water) are ignored. The only effect of the solvent
molecules is to dictate theelectrical permittivity ε of the mean field. The Boltzmann approximation leads to the
simple conclusion that the energy of an atom is given exclusively by the singlet energy e1(i)—namely,
that

(H.1)

which was presented earlier as Equation 9.1, fully describes the potential energy landscape that dictates atom
distributions. This is asinglet potential, meaning that the potential is a function of position, but not a
function of the location of or properties of other atoms. The potential energy of a system owing to singlet
energies has the mathematical form E =∑ie1(i), where E is the potential energy, i denotes atom
i, e1 is the singlet energy, and i is the position vector of atom i. These terms carry the effects of
external fields on the atoms, but do not account for atom-atom interactions. For example, if walls are at
x = 0 and x = L, e1 would be zero far from the walls, but would become large as x approached 0 or
L.

Unfortunately, as systems become more dense, we must consider the interaction of systems. This leads to pairpotentials, which are a function of two atoms and their relative location, triplet potentials, which are a function of
the relative positions of sets of three atoms, and so on. The following sections discusspair potentials in some detail.
As compared to a Boltzmann model, considering these pair potentials in detail leads to vastly different solutions for
the distribution of atoms, as shown in Figure H.1. We describe these in the coming sections with reference to the
multipolar theory in Appendix F.

Pair potentialsimply energies that are given by the interactions between pairs of atoms. These energies have the
mathematical form E =∑j>i∑ie2(i,j). The sum denoted by ∑j>i, in this context, means that the energy from the
interactions between molecules i and j are counted only once. The most fundamental challenge and limitation of
molecular dynamics is the definition and evaluation of the pair potential e2. In particular, pair potentialscan be
precisely defined only if a large amount of data (including fluid properties, quantum modeling, and spectroscopic
data) is used. They can be rather precisely defined for simple atoms (e.g., liquid Argon), but they are difficult to
define for more complex structures, and are more difficult to define for water. Even once a pair potential formula is
agreed upon, evaluating the potential numerically is computationally expensive. If all pair potentials in a
system with N molecules is considered, order N2 evaluations are required. Given that N is typically
large, this is onerous and approximate techniques must be used. When the number of evaluations is
reduced with some sort of engineering approximation (say, by only calculating energies for molecules
that are reasonably close to each other), the true pair potential no longer conserves momentum and
energy, and typically must be replaced with an effective pairpotential (or some other approximation) that
corrects for these limitations. Finally, the pair potentials are typically corrected for tripletand higher-order
energies.

H.1.1 Monopole pair potentials

The simplest pair potential considers the electrostatic interaction between the charges on atoms. Recall that the
electrostatic potential energy in a vacuum between two atoms with point charges q1 and q2 is

(H.2)

where Δr12 is the length of the vector between the two locations 1 and 2. The force is simply the derivative of the
potential with respect to Δr12:

(H.3)

If the atoms are no longer in a vacuum, we must approximate the interactions of the atoms with all of the other
atoms (i.e., the solvent atoms) somehow. Here, we often make a continuum approximation that the medium in
which the points exist can be described by a single value (the electric permittivity) which describes how the potential
is attenuated by the dielectric screening (see Section 5.1.2):

(H.4)

The only difference is that ε0 has been replaced by ε, the electrical permittivity of the medium.

H.1.2 Spherically-symmetric multipole pair potentials

All atomic interaction potentials that are not described by point charges (i.e., monopolar interactions) correspond to
distributions of charge (i.e., multipoles). The forces caused by electrostatic interactions between electron orbitals of
molecules are typically described by a time-averaged potential that accounts for all of the multipolar electrostatic
interactions. We consider only spherically-symmetric potentials, since most models for water involve a
spherically-symmetric pair potential combined with monopole pair potentials to account for the partial charge on
each of the atoms.

Hard sphere potential

A strong, short-ranged repulsion prevents multiple atoms from existing in the same location at the same
time—this stems from the Pauli exclusion principle as applied to the electron orbitals. A simple way to
handle this is to treat the atoms as hard spheres. From the standpoint of pair potentials, this corresponds
to

(H.5)

where d12 is the effective hard-sphere diameter of the two-body system. Hard-sphere systems are simple to study
and exhibit some semblance of reality, but fail to predict a number of simple phenomena, such as a liquid-gas
transition.

Lennard-Jones potential

Real atoms always show long-range Coulomb attraction known as Van der Waals attraction that scales inversely with
the sixth power of the distance. The short-range repulsive potential is large but neither infinite nor discontinuous,
and it has been deemed convenient to approximate this with a Δr12-12 dependence. There is no real physical source
for the scaling of this term, but it works well and liquid properties are not a strong function of the functional form
used for the repulsion term. More importantly, it is computationally more efficient to calculate an r-12 term since the
r-12 term is the square of the r-6 term. Thus, the most common form of pair potential has historically
been a Lennard-Jones potential (alternately called an LJ potential or LJ 6-12 potential), with general
form

where εLJ is the depthof the potential well and σLJ is the pointat which the pair potential is zero. This potential is
straightforward to define mathematically, efficient computationally, and roughly approximates real pair
potentials for spherically-symmetric atoms. An example is shown in Figure H.2. The value of σLJ
for Argon atoms is approximately 3.4 Å, and the value of εLJ∕kB for Argon atoms is approximately
121 K.

The depth of the potential well naturally leads to a means for normalizing the temperature. We define a reference
temperature εLJ∕kB, which is the temperature at which kBT is equal to the well depth. Then a nondimensional
temperature T* can be defined as

See Appendix E for further discussion on nondimensionalization. At T* smaller than one, we expect the
intermolecular potentials to strongly affect the atom distributions. At T* much larger than one, we expect the
intermolecular potentials to not affect the atom distributions much at all.